# Unit information: Introduction to Proofs and Group Theory in 2022/23

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Unit name Introduction to Proofs and Group Theory MATH10010 20 C/4 Teaching Block 4 (weeks 1-24) Dr. Mackay Not open A in A Level Mathematics or equivalent None School of Mathematics Faculty of Science

## Unit Information

Lecturers: Jos Gunns and John Mackay

Unit Aims

This unit aims to to introduce students to fundamental concepts in Mathematics including set theory, techniques of proof and group theory.

Unit Description

The first half provides an introduction to logical propositions, basic set theory and cardinality, functions and relations, and proof techniques. These notions are exemplified with some topics from elementary number theory, such as the Fundamental Theorem of Arithmetic, Euclid’s algorithm, modular arithmetic.

The second half explores the area of group theory. In the past, certain systems studied in various parts of mathematics have turned out to have common features, and these have been formalised into the definition of a group. Some of the earliest examples arose in connection with the solution of polynomial equations by formulae, and involved what we would now call groups of permutations. Other examples arise in trying to pin down mathematically what it means to say that a geometrical figure is symmetric and to quantify just how symmetric it is. It makes sense to study in one go all the systems which have the same general features. We shall start from the formal definition of a group and derive important general results from it using careful mathematical reasoning, but throughout there will be an emphasis on particular examples in which calculations can be performed relatively easily. The unit aims to introduce students to basic material in group theory, including examples of groups, group homomorphisms, subgroups, quotient groups, basic theorems on groups (such as Lagrange’s Theorem, Fermat’s Little theorem, 1st Isomorphism Theorem) and their applications.

## Your learning on this unit

At the end of the unit, the students should:

• be able to distinguish correct from incorrect and sloppy mathematical reasoning,
• be able to understand and write clear mathematical statements and proofs;
• be able to produce proofs using mathematical induction;
• be able to correctly use quantifiers and to negate logical statements that include quantifiers and connectives;
• be able to work with sets, whether they are finite, countable, or uncountable;
• be proficient in establishing fundamental properties of functions, such as injectivity and surjectivity;
• be proficient in using Euclid’s algorithm and modular arithmetic.
• be able to demonstrate facility in working with various specific examples of groups;
• be able to solve standard types of problems in introductory group theory;
• understand and be able to apply the basic concepts and results presented throughout the unit.

## How you will learn

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

## How you will be assessed

Assessment for learning/Formative assessment:

• problem sheets set by the lecturer and marked by the students' tutors.

Assessment of learning/Summative assessment:

• Two timed, open-book examinations (each worth 45%) after each teaching block
• Coursework (10%)

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH10010).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.